There are lots of good algebraic geometry textbooks available, none of which is exactly right. My official recommendation is:

on the theory that you should have something you can use as a reference for algebraic geometry.

My lectures won't follow his, but I'll try to indicate how what we're doing in class links to that textbook.

Here are some other books which I'll consult during the semester.

D. Mumford, Algebraic geometry 1: complex projective varieties, Springer-Verlag.
Philosophically, if not literally, this semester's approach to algebraic geometry will be very similar to the one Mumford uses here.
R. Hartshorne, Algebraic geometry, GTM 52, Springer-Verlag.
This is a spare ("French") account of algebraic geometry, as opposed to Shafarevich's lush ("Russian") version. This is a fantastic, if somewhat forbidding, reference.
K. Hulek, Elementary algebraic geometry, Student Mathematical Library vol. 20, AMS.
This can't be your only book on algebraic geometry, since it doesn't go far enough to serve as a reference; but what it does, it does very well.